# Exponential Functions

Exponential functions are explored, interactively, using an applet. The properties such as domain, range, horizontal asymptotes, x and y intercepts are also investigated. The conditions under which an exponential function increases or decreases are also investigated.

Sliders in the applet control panel are used to change parameters included in the definition of the exponential function which in this tutorial has the form

f(x)=a*B(b(x+c)) + d

The values of the coefficients a, b, c, d, and the base B may be changed continuously (small increments). This makes this interactive tutorial very helpful and leads to a deep understanding of the behavior of the graph of the exponential functions.

## Definition of the Exponential Function

The basic exponential function is defined by

f(x) = Bx

where B is the base such that B > 0 and B not equal to 1.
The domain of f is the set of all real numbers.

Example:

1. f(x) = 2x
2. g(x) = 4x
3. h(x) = 0.4x
4. k(x) = 0.9x

Exponential functions may be explored using an html5 applet (suitable for ipads and tablets)

or a java applet

### 2)Interactive Tutorial Using Java Applet (1)

Your browser is completely ignoring the <APPLET> tag!

1. Click on the button above "click here to start" and maximize the window obtained.
2. Use the sliders on the left panel of the applet to set a to 1, b to 1, c to 0, d to 0 and the base B to 2. This defines function f given in part a) in the example above. Zoom in and out if necessary. Read values from the graph and make sure that the graph you have corresponds to the function defined above. Does the graph of function f increase or decrease?
3. Use the sliders again to set a to 1, b to 1, c to 0, d to 0 and the base B to 4. This defines function g given in part b) in the example above. Again make sure that the graph corresponds to function g above. Does the graph of function f increase or decrease?
4. Use the sliders again to set a to 1, b to 1, c to 0, d to 0 and the base B to 0.4. This defines function h given in part c) in the example above. Again make sure that the graph corresponds to function g above. Does the graph of function f increase or decrease?
5. Use the sliders again to set a to 1, b to 1, c to 0, d to 0 and the base B to 0.9. This defines function h given in part c) in the example above. Again make sure that the graph corresponds to function k above. Does the graph of function f increase or decrease?

## Increase and Decrease of the Exponential Functions

### Interactive Tutorial Using Java Applet (2)

1. Set a to 1, b to 1 , c to 0, d to 0 and change base B so that B > 1. Note that as long as B > 1, the exponential function Bx increases throughout its domain which is the set of all real numbers.
2. Set a to 1, b to 1 , c to 0, d to 0 and change base B so that 0 < B < 1. Note that as long as 0 < B < 1, the exponential function Bx decreases throughout its domain.

## Range and Horizontal Asymptote of the Exponential Functions

### Interactive Tutorial Using Java Applet (3)

1. Use the sliders to set a to 1, b to 1, c and d to zero. Set base B values greater than 1 and note the following: as x increases, Bx increases without bound (zoom in and out if necessary) and as x decreases Bx approaches zero but is never equal to zero. The graph follows the x axis. The range of Bx is given by the interval (0 , + infinity). The x axis (y = 0) is the horizontal asymptote.
2. Use the sliders to set a to 1, b to 1, c and d to zero. Set base B to values smaller than 1 and note the following: as x decreases, Bx increases without bound (zoom in and out if necessary) and as x increases Bx approaches zero but is never equal to zero. The graph follows the x axis. The range of Bx is given by the interval (0 , + infinity). The x axis (y = 0) is the horizontal asymptote.

## Shifting, Scaling and Reflection of the Exponential Functions

We now investigate the effects of parameters a, b, c and d on the properties of the graph of function f defined by:

f(x)=a*B(b(x+c)) + d

### Interactive Tutorial Using Java Applet (4)

1. Set B = e, b = 1, c = 0 and d = 0 and Explore the effects of parameter a (vertical scaling) on the graph of f.
2. Set a = 1, c = 0, d = 0 and B = e and Explore the effects of parameter b (horizontal scaling) on the graph of f.
3. Set a = 1, b = 1, d = 0 and B = e and Explore the effects of parameter c (horizontal shift) on the graph of f.
4. set B,a,b,c to values of your choice, change d and explain how it affects the horizontal asymptote and the range of f.
5. What parameter(s) affect the y intercept? Do you think the graph of this function will always have a y intercept? Explain analytically.
6. What parameter(s) affect the x intercept? Do you think the graph of this function will always have an x intercept? Explain analytically.

You may want to work through a tutorial on finding Exponential Function Given its Graph. It is a tutorial that complements the one on this page.

More references and links to topics related to exponential functions.